$ A = \left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]$ $ C = \left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]$ Is $ A+ C$ defined?
Answer: In order for addition of two matrices to be defined, the matrices must have the same dimensions. If $ A$ is of dimension $( m \times  n)$ and $ C$ is of dimension $( p \times  q)$ , then for their sum to be defined: 1. $ m$ (number of rows in $ A$ ) must equal $ p$ (number of rows in $ C$ ) and 2. $ n$ (number of columns in $ A$ ) must equal $ q$ (number of columns in $ C$ Do $ A$ and $ C$ have the same number of rows? Yes Yes No Yes Do $ A$ and $ C$ have the same number of columns? Yes Yes No Yes Since $ A$ has the same dimensions $(2\times2)$ as $ C$ $(2\times2)$, $ A+ C$ is defined.